Sell 'em both and find a used Sako TRG-42 in .300win mag.
The action is better than any "custom", great barrel, very fine stock system and many accessories. Already threaded for a brake, and might come with one.
The VERSATILE thing about the TRG rifles is THE ACTION... With spare bolt bodies, one rifle can take any standard .476 ctg, the .533 standard magnum, and the .588 super-mag ctgs. Easy to switchbarrel. Without a scope, probably about $2400 in good used condition.
Really need a brake for a WSM ctg? Using one to beat the recoil just blinds you to the fact that heavy loads will fire-bomb your throat. A stainless barrel erodes in crocodile type gaps compared to a CM barrel that erodes by cracking. Barrel life isn't that good if you're not watching the temps your charge generates.
Neither the Browning or Tikka T3 is comparable to a built longrange rifle. Trying to make it one will cost you well over a thousand bucks, likely more than $1500 and still haven't bought a scope...
The Tikka T3 Varmint and Tactical models are excellent values for their heavy barrels (Sako quality) and integral Picatinny rail (Tactical). The Tikka competition rifle w/laminated stock at about $1600 is another "built" option; if you can live with the laminated stock. It's kind of gawdy looking, but that's why we use Krylon paint.
Another Superb Option is the FN-H Police or Patrol rifle with chrome-lined match barrel. Some were made in .300wsm if you're committed to that chambering. Actually, though... Go .308win or .260rem and learn to compensate. A used FN-H (basically an accurized match quality Win 70 Classic) is really The Sleeper of precision rifles. Comes with a Near Mfg Picatinny rail, or it should... The Police model has McMillan stock choices, the Patrol you can restock with an H-S Precision for about $300, or you might like the Hogue with full alum bedding block. Buy one for around $800 and see how it shoots before you buy a new stock...
All the recommendations of buying used are made with the understanding that you will order a new custom barrel within weeks of getting your rifle. Shoot it till the barrel is shot out and then you'll KNOW your gun. I would study the hell out of the better Loading Manuals and get some ballistics software for comparisons. The Sierra book & software pkg is a good value.